Why are an audio signal and its cochlear response modelled as two-dimensional variables? According to Eguíluz et al, amplification of an incoming audio signal by the human ear can be described by a supercritical Hopf-bifurcation
$$\dot x(t) =(\lambda-i\omega_0-|x(t)|^2)x(t)+S(t),$$
where $\lambda<0$ is the bifurcation parameter located close to the bifurcation point (i.e. $\lambda\approx0$), $\omega_0>0$ is the Hopf frequency of a particular hair cell in the cochlea, and the non-homegeneous term
$$S(t)=F\cdot(\cos \omega t+i\sin\omega t)$$
represents the incoming signal with amplitude $F>0$ and frequency $\omega\approx\omega_0$. One can show that for $F$ not too small
$$x(t)\approx F^{1/3}\cdot(\cos \omega t+i\sin\omega t).$$
So far, so good.
However, I am a bit puzzled by the fact that both the input $S$ and the response $x$ are complex variables, and hence correspond to two-dimensional real signals. I would intuitively expect a simple audio signal to be just of the real/one-dimensional form $F\cos\omega t$. Since no different frequencies or different amplitudes are involved, I do not think that this additional dimension is due to overtones. But how am I to interpret it?
 A: The equation $$\dot z = (\mu +\mathfrak{j}\omega_0)z-|z|^2z+Fe^{\mathfrak{j}\omega t} \tag {1}\label {1}$$ from Eguiluz, etc., is one of the simplest model of entrainment, that is of a phase lock "loop". 
The function $z_1(t)=A \, e^{\mathfrak{j}\omega t}$ for some $A$ is a stationary solution of $\eqref{1}$ since $|z_1|^2=|A|^2$ and $\dot z_1 = A \mathfrak{j}\omega\, e^{\mathfrak{j}\omega t}$, 
therefore 
$A\mathfrak{j}\omega e^{\mathfrak{j}\omega t}= (\mu +\mathfrak{j}\omega_0) A e^{\mathfrak{j}\omega t} -|A|^2Ae^{\mathfrak{j}\omega t}+Fe^{\mathfrak{j}\omega t}$ and upon dividing both sides with the exponential $e^{\mathfrak{j}\omega t}$ we get
$$A\mathfrak{j}\omega = A(\mu+\mathfrak{j}\omega_0 - |A|^2)  +F \tag{2}\label{2}$$
This equation $\eqref{2}$ is a complex cubic in the unknown complex amplitude $A$, and can be solved given $\mu, \omega, \omega_0$ and $F$. 
If we set the time reference so that $F$ be real then the phase of $A$ represents the phase delay of the entrained oscillation $z_1(t)=Ae^{\mathfrak{j}\omega t}$ of complex amplitude $A$ relative to that of the driving force.
